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PHYSICAL REVIEW E 90, 021001(R) (2014)
Flagella-induced transitions in the collective behavior of confined microswimmers Alan Cheng Hou Tsang and Eva Kanso* Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California 90089, USA (Received 5 May 2014; published 28 August 2014) Bacteria are usually studied in free-swimming planktonic state or in sessile biofilm state. However, little is known about intermediate states where variability in the environmental conditions and/or energy supply to the flagellar propulsive system alter flagellar activity. In this Rapid Communication, we propose an idealized physical model to investigate the effects of flagellar activity on the hydrodynamic interactions among a population of microswimmers. We show that decreasing flagellar activity induces a hydrodynamically triggered transition in confined microswimmers from turbulentlike swimming to aggregation and clustering. These results suggest that the interplay between flagellar activity and hydrodynamic interactions provides a physical mechanism for coordinating collective behaviors in confined bacteria, with potentially profound implications on processes such as molecular diffusion and transport of oxygen and nutrients that mediate transitions in the bacteria physiological state. DOI: 10.1103/PhysRevE.90.021001
PACS number(s): 47.63.Gd, 05.65.+b, 87.18.Fx, 87.18.Hf
Collective motion emerges in a wide range of natural systems, from fish schools [1,2] to bacterial colonies [3–8], and is believed to play important roles in the functioning and survival of the group. For example, many species of bacteria cyclically transition from a free-swimming planktonic state into a sessile biofilm state in response to environmental conditions [9,10]. It is well known that the free-swimming state is characterized by flagella-driven motility, which is suppressed in the biofilm state. However, the effect of intermediate states—where variability in the environmental conditions and/or energy supply to the flagellar propulsive system may alter the level of flagellar activity—on the emergent bacterial behavior remains largely unexplored. In this Rapid Communication, we show in the context of an idealized model that a decrease in flagellar activity could lead microswimmers, via hydrodynamic interactions only, to transition from a turbulentlike swimming state, akin to the one observed in numerous experiments, to clustering and aggregation. The model considers asymmetric (head-tail) microswimmers that are strongly confined in a thin film of Newtonian fluid, with the dimension of the swimmers being comparable to the thickness of the fluid film (see Fig. 1). Geometric confinement is a common feature of the natural environment of various bacteria species. It is also characteristic of several experimental setups on biological and artificial microswimmers [6,7,11,12]. Confined microswimmers have a distinct hydrodynamic signature in the sense that the far-field flow is that of a two-dimensional (2D) potential source dipole as opposed to the three-dimensional (3D) force dipole in the unbounded case [13]. Thus, the usual categorization of unbounded swimmers into pushers and pullers [14] becomes irrelevant. The dipolar far field is independent of the transport mechanism (driven particles or self-propelled swimmers) and is rooted in the fact that the basic physics in confined fluids is that of a Hele-Shaw potential flow [15]. Further, due to friction with the nearby walls, confined swimmers with geometric polarity (large head or large tail) reorient in response to both the local flow field and its gradient [13], and their
*
Corresponding author: [email protected]
1539-3755/2014/90(2)/021001(5)
collective behavior exhibits instabilities that are qualitatively distinct from those observed in unbounded swimmers [13,16]. It is also worth noting that asymmetric self-propelled particles interacting sterically, not hydrodynamically, were recently shown to exhibit rich phenomena including aggregate formation [17]. One of our main goals in this Rapid Communication is to formulate a particle swimmer model that takes into account the effect of flagellar activity. We first establish that the intensity of the dipolar far field induced by a beating flagellum depends on the level of flagellar activity: Vigorously beating flagella induce stronger dipolar fields than weakly beating ones. The gap-averaged flagellar motion is prescribed as y(x,t) = A cos(kx − t), with x ∈ [−1,1], and is assumed to induce a constant swimming velocity U in the −x direction. Here, all parameters are dimensionless with the characteristic length and time scales being set by the flagellum’s halflength and beating frequency, respectively. The potential flow perturbation induced by this beating motion is computed numerically for various values of the beating amplitude A and wavelength k while normalizing the swimming velocity U to 1. The time-averaged flow field over one beating cycle is approximated, using a standard fitting method based on minimization of the L2 norm, by the dipolar field of a circular disk with effective radius Rtail moving at the same swimming velocity U (see Fig. 2). Note that the dipolar field√induced by a circular disk located at zo = xo + iyo (i = −1) in the complex z plane and oriented at an arbitrary angle αo to the x axis can be described by the complex velocity 2 w(z) = ux − iuy = σ eiαo /(z − zo )2 , where σ = U Rtail is the dipole strength. Figure 2(c) shows that, as A and k increase, Rtail increases accordingly. Consequently, the dipole strength σ increases with increasing flagellar activity. We now model the flagellar far-field flow by that of a circular disk of effective radius Rtail and we assume that the hydrodynamic coupling between the head and the flagellum is weak. This leads to a head-tail dumbbell swimmer model, where the value of Rtail is interpreted as a measure of the flagellar activity. A derivation of the equations governing the motion of such weakly coupled dumbbell swimmer can be found in Ref. [13] and therefore is omitted here. The dynamics of a population of N such swimmers can be expressed in
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flagellar activity, and vice versa. For example, according to an experimental study on Rhizobium lupini [18], the average head size corresponds to Rhead ≈ 0.4 μm whereas depending on environmental conditions (viscosity, temperature, pH value), the flagella-beating amplitude is reported to vary between 0 and 0.4 μm. Now, assuming an average wave number of 5π , this variation in amplitude corresponds, in our model, to a variation in Rtail from 0 to 0.62 μm, thus causing a change in sign of Rhead − Rtail and hence μhead − μtail . In the model, this signed difference dictates the signed value of ν2 = (μhead − μtail )/, being the distance between the hydrodynamic centers of the head and tail, and therefore how a swimmer reorients in response to the local flow. Vigorous flagellar activity for which μhead − μtail > 0 (i.e., ν2 > 0) causes swimmers to reorient in the direction of the local flow whereas swimmers with weakly beating flagella for which μhead − μtail < 0 (i.e., ν2 < 0) reorient in the opposite direction to the local flow. Swimmers also reorient in response to the flow gradient as indicated by the ν1 term in Eq. (1), consistently with the classical Jeffery’s orbit [19]. Note that ν1 is proportional to μhead + μtail . To close the model in Eq. (1), we need to evaluate the velocity field w(z) induced by N potential dipoles in a doubly periodic domain, which involves the evaluation of conditionally convergent, doubly infinite sums of terms that decay as 1/|z|2 . In this Rapid Communication, we present a closed-form solution for this doubly periodic system. We distinguish our exact analytical solution from the approximate numerical solution in Ref. [16]. We showed in Ref. [20] that the velocity field associated with a system of finite dipoles in a doubly periodic domain can be expressed in terms of the Weierstrass zeta function and, in Ref. [21], we derived a point dipole model that is consistent with both the finite dipole system and the model in Eq. (1). Building upon these results, we get, after some straightforward but tedious manipulations, that the velocity field induced by N potential dipoles located at zn with orientation αn , n = 1, . . . ,N, in a doubly periodic domain can be written in terms of the Weierstrass elliptic function as follows:
μhead
μ tail
FIG. 2. (Color online) (a) Time-averaged flow field created by a beating flagellum with A = 0.5, k = π , and U = 1. Inset: Snapshots of the flagellum-induced flow field at different times. (b) Dipolar field created by a circular disk of effective radius Rtail fitted to the average flow field in (a). (c) Change in Rtail with A and k of the traveling wave via the flagellum. (d) Reduction of a flagellated swimmer to a dumbbell swimmer.
Here, σn is the strength of the potential dipole associated with the nth swimmer. The Weierstrass elliptic function ρ(z) is given by ρ(z; ω1 ,ω2 ) = z12 + k,l ( (z− 1 kl )2 − 12 ), with kl kl = 2kω1 + 2lω2 , k,l ∈ Z − {0}, and ω1 and ω2 being the half-periods of the doubly periodic domain. This function has infinite numbers of double pole singularities located at positions of z = 0 and z = kl , corresponding to the 1/|z|2 singularities induced by the potential dipoles. In addition to the hydrodynamic coupling, we account for steric interactions in Eq. (1) using a collision avoidance mechanism Vn based on the repulsive part of the LeonardJones potential. These near-field interactions decay rapidly outside a small excluded area centered around zn . Their rapid decay ensures that the order of the far-field hydrodynamic interactions is preserved. We focus on the evolution of populations of microswimmers that are initially randomly oriented but spatially
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FIG. 3. (Color online) (a) Swirling motion in vigorously beating flagellated swimmers; (b) orientational order; and (c) aggregation in weakly beating flagellated swimmers. Probability distributions of velocity of the swimmers are depicted in the lower row.
homogeneous. We normalize U and σn to 1 and set μ = 0.5, N = 100. We vary ν1 , ν2 , and the area fraction A = N A/A0 , where A is the area of the microswimmer, and A0 = L2 is the size of the doubly periodic square domain. We perform Monte Carlo type simulations in the sense that, for each set of parameters (ν1 ,ν2 , A ), we run multiple trials corresponding to different sets of initial conditions, each set taken from a uniform probability distribution function. We observe the emergence of three distinct types of global structures: swirling behavior for vigorously beating flagella, orientational order for a narrow range of parameter values around ν2 = 0, and aggregation or clustering for weakly beating flagella. Representative simulations are shown in Fig. 3. The swirlinglike motion is characterized by a velocity distribution function with a mean value higher than the speed of the individual swimmer [Fig. 3(a)]. This collective behavior where vortexlike structures emerge, break, and rearrange elsewhere is reminiscent of the bacterial turbulence observed in numerous experiments [3–6,8,22,23]. The associated increase in swimming speed was also observed experimentally. In the context of the dipole model, the increase in speed can be explained as follows. Swimmers with vigorously beating flagella tend to “tailgate” each other as a result of them aligning with the local flow field. When a swimmer is close to another swimmer, it will orient towards and travel along the streamlines of the potential dipole created by the nearby swimmer as depicted schematically in Fig. 4(a). As the swimmers align and form a chainlike structure, they create a flow field that helps their forward motion, thus increasing the swimmers’ velocities, as evidenced from the probability distribution function in Fig. 3(a). As the flagellar activity decreases (by decreasing the value of ν2 ), a transitional behavior that does not exhibit a clear pattern is observed before a global orientational order develops [see Fig. 3(b)]. The development of orientational order happens at a much longer time scale than the swirling dynamics and is a result of the dipoles reorienting with the local velocity gradient
(nonzero ν1 ), which is ignored in Refs. [13,16]. In this case, the velocity distribution function is Gaussian centered at U = 1 [Fig. 3(b)], implying that this collective mode has no advantage at the population level in terms of increased swimming speed. When we further decrease the flagellar activity, the swimmers begin to aggregate and cluster [see Fig. 3(c)]. The clustering behavior of swimmers with weak flagellar activity can be explained by recalling that such swimmers reorient in the opposite direction of the local flow field. Therefore, a swimmer tends to travel in the opposite direction to the streamlines created by a nearby swimmer [see Fig. 4(b)], which leads to aggregation. The collective aggregation of many swimmers takes place at a very short time scale and leads to the formation of clusters. Some clusters are unstable and break readily [blue circle in Fig. 3(c)]. However, long-lived stable clusters also form [red circle in Fig. 3(c)]. The stable cluster depicted in Fig. 3(c) slows down considerably as more swimmers join the cluster. The velocity distribution function has a strong peak around zero velocity because most swimmers are attached to the stable cluster, with the remaining swimmers moving at unit speed. We use a number of statistical measures to assess the observed global structures. In addition to the velocity distribution function shown in Fig. 3, we compute the velocity and angular correlation functions Cv and Cθ as a measure
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FIG. 4. (Color online) (a) Swimmers with vigorously beating flagella orient with local flow and thus tend to chase each other. (b) Swimmers with weakly beating flagella orient in the opposite direction to local flow and thus tend to aggregate together.
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of the spatial range for which the velocity and orientation of a swimmer are coordinated with those of its neighbors [7]. Local correlations are observed for the swirling-type motion while global correlations are seen when orientational order is developed [Fig. 5(a)]. We also compute the polar order iαn (t) parameter P (t) = N1 | N | to assess the degree of n=1 e global order in the swimmers population [Fig. 5(b)]. However, these statistical functions do not distinguish between the swirling behavior, for which the velocity and orientation are locally but not globally correlated, and the transitional behavior between the three global modes reported here, which is also characterized by locally correlated velocity and orientation but no swirling patterns. Therefore, we introduce a rotational activity parameter κ(t) = N1 N ˙ n (t)|/|˙zn (t)| n=1 |α that measures the average change in orientation weighted by the traveling distance [Fig. 5(c)]. The values of κ exhibit a continuous transition as ν2 decreases as shown in Fig. 6(a). Figure 6 also shows that the general dependence of κ and longtime developed P on ν2 and A is insensitive to variations in initial conditions. We set threshold values for κ and P , and use the mean of the velocity distribution function to distinguish between the three global modes: swirling, orientational order, and aggregation. These behaviors are mapped onto the 3D parameter space (ν1 ,ν2 , A ) in the phase diagram depicted in
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Fig. 7. Note that the orientational order mode arises in a narrow parameter range where ν2 is small and the hydrodynamic interactions are dominated by the weaker gradient term in Eq. (1), which explains why it develops at a much slower time scale. We considered the effects of rotational Brownian noise on the global modes by adding a rotational diffusion term [Re Dr ζ ieiαn ] to the orientation equation in Eq. (1). Here, we set Dr to have values of order 10−2 so that the corresponding P´eclet number Pe = (ν1 + ν2 )σ U A /(2Dr ) is of order of 1. At this order of Pe numbers, the addition of Brownian noise had negligible effects on the emergence of the swirling, orientational order, and aggregation phase, causing only small perturbations to the “boundaries” of these global modes. To conclude, we note that, whereas our finite-sized system is fully nonlinear and nondilute, the (ν2 , A ) slice of Fig. 7 may be interpreted as a rough analog to the 2D linear stability diagram obtained in the kinetic model of [13, Fig. 2], where ν1 was considered identically zero. The linear instabilities reported in the kinetic model do not reveal the nature of the emergent collective modes (swirling versus clustering) nor their physical implications. Here, we showed that transitions from swirling to clustering and aggregation, and vice versa, can be induced by an interplay between the flagellar activity
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FIG. 7. (Color online) Phase diagrams showing the three different collective modes swirling, orientational order, and aggregation, as a function of ν1 , ν2 , and A . Regions of transitional behavior for which no clear pattern is identified are left in white.
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and the hydrodynamic interactions. These results, albeit in the context of a simplified model, may have profound implications on understanding intermediate stages where flagellar activity is decreased in response to environmental conditions such as overcrowding or nutrient depletion. Our findings suggest that
the interplay between hydrodynamics and flagellar activity may serve as a physical mechanism for aggregating the bacteria, thus altering their convective flow which may, in turn, alter microbial processes such as transport of oxygen and nutrients.
[1] I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, J. Theor. Biol. 218, 1 (2002). [2] I. D. Couzin, J. Krause, N. R. Franks, and S. A. Levin, Nature (London) 433, 513 (2005). [3] L. H. Cisneros, R. Cortez, C. Dombrowski, R. E. Goldstein, and J. O. Kessler, Exp. Fluids 43, 737 (2007). [4] A. Sokolov, I. S. Aranson, J. O. Kessler, and R. E. Goldstein, Phys. Rev. Lett. 98, 158102 (2007). [5] A. Sokolov, R. E. Goldstein, F. I. Feldchtein, and I. S. Aranson, Phys. Rev. E 80, 031903 (2009). [6] N. C. Darnton, L. Turner, S. Rojevsky, and H. C. Berg, Biophys. J. 98, 2082 (2010). [7] H. P. Zhang, A. Be’er, E.-L. Florin, and H. L. Swinney, Proc. Natl. Acad. Sci. USA 107, 13626 (2010). [8] J. Dunkel, S. Heidenreich, K. Drescher, H. H. Wensink, M. B¨ar, and R. E. Goldstein, Phys. Rev. Lett. 110, 228102 (2013). [9] M. B. Miller and B. L. Bassler, Ann. Rev. Microbiol. 55, 165 (2001). [10] A. C. Edmunds, L. F. Castiblanco, G. W. Sundin, and C. M. Waters, J. Bacteriol. 195, 2155 (2013). [11] S. Thutupalli, R. Seemann, and S. Herminghaus, New J. Phys. 13, 073021 (2011).
[12] A. Bricard, J.-B.Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Nature (London) 503, 95 (2013). [13] T. Brotto, J.-B. Caussin, E. Lauga, and D. Bartolo, Phys. Rev. Lett. 110, 038101 (2013). [14] D. Saintillan and M. J. Shelley, Phys. Rev. Lett. 100, 178103 (2008). [15] T. Beatus, R. H. Bar-Ziv, and T. Tlusty, Phys. Rep. 516, 103 (2012). [16] A. Lefauve and D. Saintillan, Phys. Rev. E 89, 021002 (2014). [17] H. H. Wensink, V. Kantsler, R. E. Goldstein, and J. Dunkel, Phys. Rev. E 89, 010302 (2014). [18] B. Scharf, J. Bacteriol. 184, 5979 (2002). [19] G. B. Jeffery, Proc. R. Soc. London, Ser. A 102, 161 (1922). [20] A. C. H. Tsang and E. Kanso, J. Nonlinear Sci. 23, 971 (2013). [21] E. Kanso and A. C. H. Tsang, Fluid Dyn. Res. (to be published). [22] N. H. Mendelson, A. Bourque, K. Wilkening, K. R. Anderson, and J. C. Watkins, J. Bacteriol. 181, 600 (1999). [23] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, Phys. Rev. Lett. 93, 098103 (2004).
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Flagella-induced transitions in the collective behavior of confined microswimmers Alan Cheng Hou Tsang and Eva Kanso* Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California 90089, USA (Received 5 May 2014; published 28 August 2014) Bacteria are usually studied in free-swimming planktonic state or in sessile biofilm state. However, little is known about intermediate states where variability in the environmental conditions and/or energy supply to the flagellar propulsive system alter flagellar activity. In this Rapid Communication, we propose an idealized physical model to investigate the effects of flagellar activity on the hydrodynamic interactions among a population of microswimmers. We show that decreasing flagellar activity induces a hydrodynamically triggered transition in confined microswimmers from turbulentlike swimming to aggregation and clustering. These results suggest that the interplay between flagellar activity and hydrodynamic interactions provides a physical mechanism for coordinating collective behaviors in confined bacteria, with potentially profound implications on processes such as molecular diffusion and transport of oxygen and nutrients that mediate transitions in the bacteria physiological state. DOI: 10.1103/PhysRevE.90.021001
PACS number(s): 47.63.Gd, 05.65.+b, 87.18.Fx, 87.18.Hf
Collective motion emerges in a wide range of natural systems, from fish schools [1,2] to bacterial colonies [3–8], and is believed to play important roles in the functioning and survival of the group. For example, many species of bacteria cyclically transition from a free-swimming planktonic state into a sessile biofilm state in response to environmental conditions [9,10]. It is well known that the free-swimming state is characterized by flagella-driven motility, which is suppressed in the biofilm state. However, the effect of intermediate states—where variability in the environmental conditions and/or energy supply to the flagellar propulsive system may alter the level of flagellar activity—on the emergent bacterial behavior remains largely unexplored. In this Rapid Communication, we show in the context of an idealized model that a decrease in flagellar activity could lead microswimmers, via hydrodynamic interactions only, to transition from a turbulentlike swimming state, akin to the one observed in numerous experiments, to clustering and aggregation. The model considers asymmetric (head-tail) microswimmers that are strongly confined in a thin film of Newtonian fluid, with the dimension of the swimmers being comparable to the thickness of the fluid film (see Fig. 1). Geometric confinement is a common feature of the natural environment of various bacteria species. It is also characteristic of several experimental setups on biological and artificial microswimmers [6,7,11,12]. Confined microswimmers have a distinct hydrodynamic signature in the sense that the far-field flow is that of a two-dimensional (2D) potential source dipole as opposed to the three-dimensional (3D) force dipole in the unbounded case [13]. Thus, the usual categorization of unbounded swimmers into pushers and pullers [14] becomes irrelevant. The dipolar far field is independent of the transport mechanism (driven particles or self-propelled swimmers) and is rooted in the fact that the basic physics in confined fluids is that of a Hele-Shaw potential flow [15]. Further, due to friction with the nearby walls, confined swimmers with geometric polarity (large head or large tail) reorient in response to both the local flow field and its gradient [13], and their
*
Corresponding author: [email protected]
1539-3755/2014/90(2)/021001(5)
collective behavior exhibits instabilities that are qualitatively distinct from those observed in unbounded swimmers [13,16]. It is also worth noting that asymmetric self-propelled particles interacting sterically, not hydrodynamically, were recently shown to exhibit rich phenomena including aggregate formation [17]. One of our main goals in this Rapid Communication is to formulate a particle swimmer model that takes into account the effect of flagellar activity. We first establish that the intensity of the dipolar far field induced by a beating flagellum depends on the level of flagellar activity: Vigorously beating flagella induce stronger dipolar fields than weakly beating ones. The gap-averaged flagellar motion is prescribed as y(x,t) = A cos(kx − t), with x ∈ [−1,1], and is assumed to induce a constant swimming velocity U in the −x direction. Here, all parameters are dimensionless with the characteristic length and time scales being set by the flagellum’s halflength and beating frequency, respectively. The potential flow perturbation induced by this beating motion is computed numerically for various values of the beating amplitude A and wavelength k while normalizing the swimming velocity U to 1. The time-averaged flow field over one beating cycle is approximated, using a standard fitting method based on minimization of the L2 norm, by the dipolar field of a circular disk with effective radius Rtail moving at the same swimming velocity U (see Fig. 2). Note that the dipolar field√induced by a circular disk located at zo = xo + iyo (i = −1) in the complex z plane and oriented at an arbitrary angle αo to the x axis can be described by the complex velocity 2 w(z) = ux − iuy = σ eiαo /(z − zo )2 , where σ = U Rtail is the dipole strength. Figure 2(c) shows that, as A and k increase, Rtail increases accordingly. Consequently, the dipole strength σ increases with increasing flagellar activity. We now model the flagellar far-field flow by that of a circular disk of effective radius Rtail and we assume that the hydrodynamic coupling between the head and the flagellum is weak. This leads to a head-tail dumbbell swimmer model, where the value of Rtail is interpreted as a measure of the flagellar activity. A derivation of the equations governing the motion of such weakly coupled dumbbell swimmer can be found in Ref. [13] and therefore is omitted here. The dynamics of a population of N such swimmers can be expressed in
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©2014 American Physical Society
RAPID COMMUNICATIONS
ALAN CHENG HOU TSANG AND EVA KANSO
PHYSICAL REVIEW E 90, 021001(R) (2014)
h
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α
y
L
x FIG. 1. (Color online) Strongly confined microswimmers create potential dipolar far-field flows.
concise complex notation z˙ n = U e−iαn + μw(zn ) + Vn , dw 2iαn ie α˙ n = Re ν1 + ν2 wieiαn . dz
(1)
Here, zn and αn denote the position and orientation of each swimmer (n = 1, . . . ,N), Re denotes the real part of the expression in brackets, whereas μ, ν1 , and ν2 are nondimensional translational and rotational mobility coefficients whose values depend on the translational mobility coefficients μhead and μtail at the swimmer’s head and tail. A swimmer can have different mobility coefficients μhead and μtail , with μtail being a decreasing function of Rtail . Given that μhead is constant, the value of μhead − μtail can thus change sign from positive to negative by decreasing the level of (b)
(a) 8
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2
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k(π)
4
5
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σn ρ(z − zn ; ω1 ,ω2 )eiαn .
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flagellar activity, and vice versa. For example, according to an experimental study on Rhizobium lupini [18], the average head size corresponds to Rhead ≈ 0.4 μm whereas depending on environmental conditions (viscosity, temperature, pH value), the flagella-beating amplitude is reported to vary between 0 and 0.4 μm. Now, assuming an average wave number of 5π , this variation in amplitude corresponds, in our model, to a variation in Rtail from 0 to 0.62 μm, thus causing a change in sign of Rhead − Rtail and hence μhead − μtail . In the model, this signed difference dictates the signed value of ν2 = (μhead − μtail )/, being the distance between the hydrodynamic centers of the head and tail, and therefore how a swimmer reorients in response to the local flow. Vigorous flagellar activity for which μhead − μtail > 0 (i.e., ν2 > 0) causes swimmers to reorient in the direction of the local flow whereas swimmers with weakly beating flagella for which μhead − μtail < 0 (i.e., ν2 < 0) reorient in the opposite direction to the local flow. Swimmers also reorient in response to the flow gradient as indicated by the ν1 term in Eq. (1), consistently with the classical Jeffery’s orbit [19]. Note that ν1 is proportional to μhead + μtail . To close the model in Eq. (1), we need to evaluate the velocity field w(z) induced by N potential dipoles in a doubly periodic domain, which involves the evaluation of conditionally convergent, doubly infinite sums of terms that decay as 1/|z|2 . In this Rapid Communication, we present a closed-form solution for this doubly periodic system. We distinguish our exact analytical solution from the approximate numerical solution in Ref. [16]. We showed in Ref. [20] that the velocity field associated with a system of finite dipoles in a doubly periodic domain can be expressed in terms of the Weierstrass zeta function and, in Ref. [21], we derived a point dipole model that is consistent with both the finite dipole system and the model in Eq. (1). Building upon these results, we get, after some straightforward but tedious manipulations, that the velocity field induced by N potential dipoles located at zn with orientation αn , n = 1, . . . ,N, in a doubly periodic domain can be written in terms of the Weierstrass elliptic function as follows:
μhead
μ tail
FIG. 2. (Color online) (a) Time-averaged flow field created by a beating flagellum with A = 0.5, k = π , and U = 1. Inset: Snapshots of the flagellum-induced flow field at different times. (b) Dipolar field created by a circular disk of effective radius Rtail fitted to the average flow field in (a). (c) Change in Rtail with A and k of the traveling wave via the flagellum. (d) Reduction of a flagellated swimmer to a dumbbell swimmer.
Here, σn is the strength of the potential dipole associated with the nth swimmer. The Weierstrass elliptic function ρ(z) is given by ρ(z; ω1 ,ω2 ) = z12 + k,l ( (z− 1 kl )2 − 12 ), with kl kl = 2kω1 + 2lω2 , k,l ∈ Z − {0}, and ω1 and ω2 being the half-periods of the doubly periodic domain. This function has infinite numbers of double pole singularities located at positions of z = 0 and z = kl , corresponding to the 1/|z|2 singularities induced by the potential dipoles. In addition to the hydrodynamic coupling, we account for steric interactions in Eq. (1) using a collision avoidance mechanism Vn based on the repulsive part of the LeonardJones potential. These near-field interactions decay rapidly outside a small excluded area centered around zn . Their rapid decay ensures that the order of the far-field hydrodynamic interactions is preserved. We focus on the evolution of populations of microswimmers that are initially randomly oriented but spatially
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FIG. 3. (Color online) (a) Swirling motion in vigorously beating flagellated swimmers; (b) orientational order; and (c) aggregation in weakly beating flagellated swimmers. Probability distributions of velocity of the swimmers are depicted in the lower row.
homogeneous. We normalize U and σn to 1 and set μ = 0.5, N = 100. We vary ν1 , ν2 , and the area fraction A = N A/A0 , where A is the area of the microswimmer, and A0 = L2 is the size of the doubly periodic square domain. We perform Monte Carlo type simulations in the sense that, for each set of parameters (ν1 ,ν2 , A ), we run multiple trials corresponding to different sets of initial conditions, each set taken from a uniform probability distribution function. We observe the emergence of three distinct types of global structures: swirling behavior for vigorously beating flagella, orientational order for a narrow range of parameter values around ν2 = 0, and aggregation or clustering for weakly beating flagella. Representative simulations are shown in Fig. 3. The swirlinglike motion is characterized by a velocity distribution function with a mean value higher than the speed of the individual swimmer [Fig. 3(a)]. This collective behavior where vortexlike structures emerge, break, and rearrange elsewhere is reminiscent of the bacterial turbulence observed in numerous experiments [3–6,8,22,23]. The associated increase in swimming speed was also observed experimentally. In the context of the dipole model, the increase in speed can be explained as follows. Swimmers with vigorously beating flagella tend to “tailgate” each other as a result of them aligning with the local flow field. When a swimmer is close to another swimmer, it will orient towards and travel along the streamlines of the potential dipole created by the nearby swimmer as depicted schematically in Fig. 4(a). As the swimmers align and form a chainlike structure, they create a flow field that helps their forward motion, thus increasing the swimmers’ velocities, as evidenced from the probability distribution function in Fig. 3(a). As the flagellar activity decreases (by decreasing the value of ν2 ), a transitional behavior that does not exhibit a clear pattern is observed before a global orientational order develops [see Fig. 3(b)]. The development of orientational order happens at a much longer time scale than the swirling dynamics and is a result of the dipoles reorienting with the local velocity gradient
(nonzero ν1 ), which is ignored in Refs. [13,16]. In this case, the velocity distribution function is Gaussian centered at U = 1 [Fig. 3(b)], implying that this collective mode has no advantage at the population level in terms of increased swimming speed. When we further decrease the flagellar activity, the swimmers begin to aggregate and cluster [see Fig. 3(c)]. The clustering behavior of swimmers with weak flagellar activity can be explained by recalling that such swimmers reorient in the opposite direction of the local flow field. Therefore, a swimmer tends to travel in the opposite direction to the streamlines created by a nearby swimmer [see Fig. 4(b)], which leads to aggregation. The collective aggregation of many swimmers takes place at a very short time scale and leads to the formation of clusters. Some clusters are unstable and break readily [blue circle in Fig. 3(c)]. However, long-lived stable clusters also form [red circle in Fig. 3(c)]. The stable cluster depicted in Fig. 3(c) slows down considerably as more swimmers join the cluster. The velocity distribution function has a strong peak around zero velocity because most swimmers are attached to the stable cluster, with the remaining swimmers moving at unit speed. We use a number of statistical measures to assess the observed global structures. In addition to the velocity distribution function shown in Fig. 3, we compute the velocity and angular correlation functions Cv and Cθ as a measure
(a)
(b)
FIG. 4. (Color online) (a) Swimmers with vigorously beating flagella orient with local flow and thus tend to chase each other. (b) Swimmers with weakly beating flagella orient in the opposite direction to local flow and thus tend to aggregate together.
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0.4 0.2 0
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FIG. 5. (Color online) Statistics associated with swirling (top row) and orientational order (bottom row): polar order parameter P , velocity and angular correlations Cv ,Cθ , and probability distribution of rotational activity parameter κ, all computed based on data corresponding to the shaded time range.
of the spatial range for which the velocity and orientation of a swimmer are coordinated with those of its neighbors [7]. Local correlations are observed for the swirling-type motion while global correlations are seen when orientational order is developed [Fig. 5(a)]. We also compute the polar order iαn (t) parameter P (t) = N1 | N | to assess the degree of n=1 e global order in the swimmers population [Fig. 5(b)]. However, these statistical functions do not distinguish between the swirling behavior, for which the velocity and orientation are locally but not globally correlated, and the transitional behavior between the three global modes reported here, which is also characterized by locally correlated velocity and orientation but no swirling patterns. Therefore, we introduce a rotational activity parameter κ(t) = N1 N ˙ n (t)|/|˙zn (t)| n=1 |α that measures the average change in orientation weighted by the traveling distance [Fig. 5(c)]. The values of κ exhibit a continuous transition as ν2 decreases as shown in Fig. 6(a). Figure 6 also shows that the general dependence of κ and longtime developed P on ν2 and A is insensitive to variations in initial conditions. We set threshold values for κ and P , and use the mean of the velocity distribution function to distinguish between the three global modes: swirling, orientational order, and aggregation. These behaviors are mapped onto the 3D parameter space (ν1 ,ν2 , A ) in the phase diagram depicted in
1= 0
κ
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FIG. 6. (Color online) (a) The time-averaged mean and standard deviation of κ and (b) the sample-averaged mean and standard deviation of long-time developed P over various initial conditions.
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0 -1 0.3
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〈P〉 t → ∞
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Fig. 7. Note that the orientational order mode arises in a narrow parameter range where ν2 is small and the hydrodynamic interactions are dominated by the weaker gradient term in Eq. (1), which explains why it develops at a much slower time scale. We considered the effects of rotational Brownian noise on the global modes by adding a rotational diffusion term [Re Dr ζ ieiαn ] to the orientation equation in Eq. (1). Here, we set Dr to have values of order 10−2 so that the corresponding P´eclet number Pe = (ν1 + ν2 )σ U A /(2Dr ) is of order of 1. At this order of Pe numbers, the addition of Brownian noise had negligible effects on the emergence of the swirling, orientational order, and aggregation phase, causing only small perturbations to the “boundaries” of these global modes. To conclude, we note that, whereas our finite-sized system is fully nonlinear and nondilute, the (ν2 , A ) slice of Fig. 7 may be interpreted as a rough analog to the 2D linear stability diagram obtained in the kinetic model of [13, Fig. 2], where ν1 was considered identically zero. The linear instabilities reported in the kinetic model do not reveal the nature of the emergent collective modes (swirling versus clustering) nor their physical implications. Here, we showed that transitions from swirling to clustering and aggregation, and vice versa, can be induced by an interplay between the flagellar activity
0.1 0 0.1
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FIG. 7. (Color online) Phase diagrams showing the three different collective modes swirling, orientational order, and aggregation, as a function of ν1 , ν2 , and A . Regions of transitional behavior for which no clear pattern is identified are left in white.
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and the hydrodynamic interactions. These results, albeit in the context of a simplified model, may have profound implications on understanding intermediate stages where flagellar activity is decreased in response to environmental conditions such as overcrowding or nutrient depletion. Our findings suggest that
the interplay between hydrodynamics and flagellar activity may serve as a physical mechanism for aggregating the bacteria, thus altering their convective flow which may, in turn, alter microbial processes such as transport of oxygen and nutrients.
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